3.918 \(\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx\)

Optimal. Leaf size=25 \[ \frac{i a (c-i c \tan (e+f x))^4}{4 f} \]

[Out]

((I/4)*a*(c - I*c*Tan[e + f*x])^4)/f

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Rubi [A]  time = 0.0676489, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ \frac{i a (c-i c \tan (e+f x))^4}{4 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]

[Out]

((I/4)*a*(c - I*c*Tan[e + f*x])^4)/f

Rule 3522

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] &&  !(IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n]))

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx &=(a c) \int \sec ^2(e+f x) (c-i c \tan (e+f x))^3 \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int (c+x)^3 \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac{i a (c-i c \tan (e+f x))^4}{4 f}\\ \end{align*}

Mathematica [B]  time = 1.40167, size = 85, normalized size = 3.4 \[ \frac{a c^4 \sec (e) \sec ^4(e+f x) (2 \sin (e+2 f x)-2 \sin (3 e+2 f x)+\sin (3 e+4 f x)-2 i \cos (e+2 f x)-2 i \cos (3 e+2 f x)-3 \sin (e)-3 i \cos (e))}{4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]

[Out]

(a*c^4*Sec[e]*Sec[e + f*x]^4*((-3*I)*Cos[e] - (2*I)*Cos[e + 2*f*x] - (2*I)*Cos[3*e + 2*f*x] - 3*Sin[e] + 2*Sin
[e + 2*f*x] - 2*Sin[3*e + 2*f*x] + Sin[3*e + 4*f*x]))/(4*f)

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Maple [B]  time = 0.003, size = 48, normalized size = 1.9 \begin{align*}{\frac{a{c}^{4} \left ( \tan \left ( fx+e \right ) +{\frac{i}{4}} \left ( \tan \left ( fx+e \right ) \right ) ^{4}- \left ( \tan \left ( fx+e \right ) \right ) ^{3}-{\frac{3\,i}{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x)

[Out]

1/f*a*c^4*(tan(f*x+e)+1/4*I*tan(f*x+e)^4-tan(f*x+e)^3-3/2*I*tan(f*x+e)^2)

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Maxima [B]  time = 1.68881, size = 81, normalized size = 3.24 \begin{align*} \frac{i \, a c^{4} \tan \left (f x + e\right )^{4} - 4 \, a c^{4} \tan \left (f x + e\right )^{3} - 6 i \, a c^{4} \tan \left (f x + e\right )^{2} + 4 \, a c^{4} \tan \left (f x + e\right )}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

1/4*(I*a*c^4*tan(f*x + e)^4 - 4*a*c^4*tan(f*x + e)^3 - 6*I*a*c^4*tan(f*x + e)^2 + 4*a*c^4*tan(f*x + e))/f

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Fricas [B]  time = 1.33357, size = 158, normalized size = 6.32 \begin{align*} \frac{4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

4*I*a*c^4/(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e)
+ f)

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Sympy [B]  time = 2.26796, size = 83, normalized size = 3.32 \begin{align*} \frac{4 i a c^{4} e^{- 8 i e}}{f \left (e^{8 i f x} + 4 e^{- 2 i e} e^{6 i f x} + 6 e^{- 4 i e} e^{4 i f x} + 4 e^{- 6 i e} e^{2 i f x} + e^{- 8 i e}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))**4,x)

[Out]

4*I*a*c**4*exp(-8*I*e)/(f*(exp(8*I*f*x) + 4*exp(-2*I*e)*exp(6*I*f*x) + 6*exp(-4*I*e)*exp(4*I*f*x) + 4*exp(-6*I
*e)*exp(2*I*f*x) + exp(-8*I*e)))

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Giac [B]  time = 1.68332, size = 82, normalized size = 3.28 \begin{align*} \frac{4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

4*I*a*c^4/(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e)
+ f)